A Surjective Characterization of Dugundji Spaces

Abstract

Abstract. It is shown that the class of Dugundji spaces coincides with the class of continuous images of the generalized Cantor set by maps satisfying the zero-dimensional lifting property. It follows that each point in a Dugundji space has a neighbourhood base of Dugundji spaces.Introduction. In [6] Pelczyñski introduced the classes of Dugundji spaces and Milutin spaces. The fact that these classes coincide for compact spaces of weight not exceeding wx was established in the joint paper of Ditor and Haydon [2]. A major breakthrough in the characterization of Dugundji spaces and Milutin spaces was achieved by R. Haydon [4] showing that every Dugundji space is a Milutin space and that the notions of Dugundji space and absolute extensor for compact zero-dimensional spaces (AE(O-dim)) are equivalent. Recently in his paper [7] Shchepin gave another characterization of the class of Dugundji spaces and produced an example of a zero-dimensional space of weight w2 that is a non-Dugundji Milutin space.In this paper we shall consider continuous surjections satisfying the zero-dimensional lifting property (z.d.l.p.) which will lead to a new characterization of Dugundji spaces. At the same time our method yields topological information about surjective maps allowing a regular averaging operator (r.a.o.) and about the local structure of Dugundji spaces.In his monograph Pelczyñski asked for a topological characterization of continuous surjections of compact metrizable spaces that allow an r

Range-preserving AE(0)-spaces

“…In Corollary 2.7 we note that a number of familiar spaces are in the class AE(0) rp , and in Corollary 2.8 we show that spaces which are "locally in AE(0) rp " are in fact in AE(0) rp . (That result is in parallel with the theorem from [14] that "locally Dugundji" implies Dugundji; the converse to that result is given by Hoffmann [11]. )…”

Linear operators with compact supports, probability measures and Milyutin maps

“…A map f : X → Y is said to be 0-invertible [20] if for any space Z with dim Z = 0 and any map p : Z → Y there exists a map q : Z → X such that f • q = p. Here, dim Z = 0 means that dim βZ = 0. We say that f : X → Y has a metrizable kernel if there exists a metrizable space M and an embedding…”